A conventional box door lock includes a deadbolt and a latch bolt. The latch bolt is normally driven by inner and outer handles, a key-operated lock unit, or a rotary knob to move a latch head of the latch bolt from a latching position to an unlatching position, such that a door can be opened. However, when a user operates the outer handle by pressing downward the same, only the latch bolt can be driven to move to the unlatching position from the latching position. To actuate the deadbolt to move to a locking position, a key is required to operate the door lock thereby causing inconvenience to the user.